What is the value of $\dfrac{d}{dx}\left(x^{-6}\right)$ at $x=1$ ?
Answer: Let's first find the expression for $\dfrac{d}{dx}\left(x^{-6}\right)$ and then evaluate it at $x=1$. The derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a negative number.) $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{-6}\right) \\\\ &=-6x^{-6-1} \gray{\text{The power rule}} \\\\ &=-6x^{-7} \end{aligned}$ So we found that $\dfrac{d}{dx}\left(x^{-6}\right)=-6x^{-7}$, which can also be written as $-\dfrac{6}{x^7}$. Now let's plug ${x=1}$ : $\begin{aligned} -\dfrac{6}{({1})^7}&=-\dfrac{6}{1} \\\\ &=-6 \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\left(x^{-6}\right)$ at $x=1$ is $-6$.